Connecting Renaissance Linear Perspective and Cartesian Geometry and Optics

Piero della Francesca View of an Ideal City (1470)

     Alberti and Brunelleschi, the founders of linear perspective in painting, "made accessible to vision . . . the modern, systematic conception of space, in a concrete artistic sphere even before abstract mathematical science had given form and force to it as a postulate."1    Brunelleschi's costruzione legittima, his method of correct or exact perspective, is concerned with representing objects themselves in proper scale and relation to one another.  Yet, from the concern for scale, Renaissance painters will develop a geometric space complimentary to the mathematical space of Descartes's philosophy.  Perspective in painting projects a plane onto its object of study and creates a one-to-one correspondence between points on the plane and points on the canvas.  Brunelleschi begins by using architectural figures such as buildings, ceilings, and tiled floors which easily match the grid structure of the projective plane.  Later, other objects will be fitted and shaped within the geometrical patterning of linear perspective.2

     In his famous essay "Perspective as Symbolic Form," Panofsky highlights the break made through linear perspective by contrasting Renaissance painting with that of Greek and Medieval works.   Prior to the Renaissance, painting concerned itself with individual objects, but the space which they inhabited failed to embrace or dissolve the opposition between bodies.  Space acted as a simple superposition, a still unsystematic overlapping.  With linear perspective comes an abstract spatial system capable of ordering objects:

As various as antique theories of space were, none of them succeeded in defining space as a system of simple relationships between height, width and depth.  In that case, in the guise of a 'coordinate system,' the difference between 'front' and 'back,' 'here' and 'there,' 'body' and 'nonbody' would have resolved into the higher and more abstract concept of three-dimensional extensions, or even, as Arnold Geulincx puts it, the concept of a 'corpus generaliter sumptum' ('body taken in a general sense'). (Panofsky 43-44)
The interest in a coordinate system and in bodies in a "general sense" developed through linear perspective corresponds to Descartes founding of a coordinate system for mathematics and his translation of corporeal bodies into geometric figures.

     With Descartes, space moves from a concern about situating and representing objects to a thinking of the spacing of space, that is how we think about the space in which subjects and objects come into being.  An understanding of Cartesian space begins with an understanding of his innovations in mathematics.  Descartes can be attributed with abstracting numbers by freeing them from spatial relation.4   From the Greeks to the scholastic period, numbers act as signs in reference to objects; numbers are a property of matter and not objects in themselves.  A number is a unit length and refers to length of a line, area, or volume.  Descartes eliminates the need for numbers to relate to things.  He thinks of numbers algebraically in which numbers are operations of relational terms.  In his Geometry Descartes uses arithmetic procedures in algebra to solve geometric problems and  vice versa.  While for the ancients, geometry was a means of solving particular problems, for Descartes, algebraic geometry constructs abstract relations without the need for a relata. 

     Descartes's work in mathematics will bring us closer to his understanding of natural philosophy since much of his concern with number and shape leads toward his investigations in mechanistic physics.  In his natural philosophy, all bodies are of one of two classes, those that have extension, corporeal substances, or those of thought, ideas.  Corporeal substances are not the bodies of everyday experience but rather geometrical objects devoid of color, texture, and smell which are secondary attributes dependent upon our experience.  It is only extension with its accompanying geometrical qualities that are necessarily a part of substances.  Consequently, "Cartesian bodies are just the objects of geometry made real, purely geometrical objects that exist outside of the minds that conceive them."5

      In order for our perceptions of the world around us to move from confused forms to clear and distinct ideas which have the clarity of mathematical objects, corporeal bodies must be interpreted by the mind with its innate geometry.6   For example, in his Optics, Descartes explains that we judge the distance of an object by calculating the distance between our two eyes and the angle formed from each eye to the object observed.  This triangulation "happens without our reflecting upon it . . . as if by a natural geometry."7   While the eyes provide a means of sight, "it is the soul which sees, and not the eyes" (Descartes 68).  Descartes maintains a fundamentally mechanistic world view in which corporeal bodies are geometric entities that fit the rational mind's natural geometry.

     Panofsky develops the correlation of Renaissance linear perspective and Cartesian philosophy by explaining that each construct the same space from different modes of thought, "'aesthetic space' and 'theoretical space' recast perceptual space in the guise of one and the same sensation: in one case that sensation is visually symbolized, in the other it appears in logical form" (Panofsky 45).  According to Jay Bolter and Richard Grusin, this new perceptual space is meant to provide an immediacy, a sense of presence with the objects represented while erasing the viewer's attention to the technique of representation.   While painting according to costruzione legittima or Alberti's Della Pittura is a painstaking process, the artist wants the viewer to forget the projective plane, the lines, and the pin holes and lose him or herself in the final product as if looking through a window onto the world.   Panofsky is aware of both a mathematical space which in fact distances the view from the object through geometry's abstraction and creates an immediacy by the psychological erasure of this distancing:
Perspective creates distance between human beings and things ('the first is the eye that sees, the second is the object seen, the third is the distance between them' says Durer after Piero della Francesca); but then in turn it abolishes this distance by, in a sense, drawing this world of things, an autonomous world confronting the individual, into the eye.  Perspective subjects the artistic phenomenon to stable and even mathematically exact rules, but on the other hand, makes the phenomenon contingent upon human beings, indeed upon the individual: for these rules refer to the psychological and physical conditions of the visual impression, and the way they take effect is determined by the freely chosen position of a subjective 'point of view.' (67)

     While Panofsky is careful to balance the artist's rules of construction with the viewer's freedom of choice in how the technique "take[s] effect," nonetheless, we are so accustomed to linear perspective that there seems little freedom to not see such objects as three dimensional representations and feel the objects are closer to us by looking through the window the artist has made. However, there is a danger in what is erased by the naturalization of Cartesian persepectivalism.  We overlook the geometrical abstractness of the objects represented, the assumption of a stationary point of view, and the construction of the viewing subject.  It is the geometric abstractness which makes cartography translatable into the picturesque.  Both partake of similar assumptions about how the subject interacts with the world and how the viewer maps space.  The naturalization of Cartesian perspectivialism along with all that it assumes in its mechanical physics affects our relation to the land.  The spacing of space founded by linear perspective and Cartesian philosophy facilitate technology's transformation of nature into an object to be possessed.  Ownership by means of seeing and representing produces the "setting-before" by which, for Heidegger, nature becomes a standing reserve, an object for human consumption.

1. Hubert Damisch. The Origin of Perspective. Cambridge, MA: MIT Press, 1995. 82-83.

2. See for example Masacio's The Trinity 1426-27 as described by Damisch (94-96).

3. Erwin Panofsky.  Perspective as Symbolic Form. New York: Zone Books, 1991. 41-43.

4. Until Descartes, mathematics relied on Aristotle's definition of number as laid out in his Metaphysics.  According to Aristotle, just as objects in the world have sensate matter, mathematical objects have noetic matter.  While sensible numbers and shapes are properties of sensible matter, noetic numbers and shapes are properties of noetic matter ( Stephen Gaukroger. "The Nature of Abstract Reasoning: Philosophical Aspects of Descartes' Work in Algebra." The Cambridge Companion to Descartes. Ed. John Cottingham. Cambridge: Cambridge UP, 1992. 98-101).

5.  Daniel Garber. "Descartes' Physics." The Cambridge Companion to Descartes. Ed. John Cottingham. Cambridge: Cambridge UP, 1992. 294.

6.  Martin Jay. Downcast Eyes. Berkeley: U California Press, 1993. 78.

7.  Rene Descartes. Optics. Descartes: Selected Philosophical Writings. Trans. John Cottingham and Robert Stoothoff. Cambridge: Cambridge UP, 1988. 67.

8. Jay David Bolter and Richard Grusin. Remediation: Understanding New Media. Cambridge, MA: MIT Press, 1999. 24-25.